Algorithmic properties of inverse monoids with hyperbolic and tree-like Sch\"utzenberger graphs

TL;DR

This paper investigates the algorithmic properties of inverse monoids with Sch"utzenberger graphs that are either tree-like or hyperbolic, showing solvability of the word problem in some cases and unsolvability in others.

Contribution

It establishes conditions under which inverse monoids have solvable word problems and characterizes the language complexity of their automata.

Findings

Finitely presented inverse monoids with tree-like Sch"utzenberger graphs have solvable word problems.

Their Sch"utzenberger automata languages are context-free.

Some inverse monoids with hyperbolic Sch"utzenberger graphs have unsolvable word problems.

Abstract

We prove that the class of finitely presented inverse monoids whose Sch\"utzenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Sch\"utzenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Sch\"utzenberger graphs and an unsolvable word problem.